Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,26)(15,22)(19,30)(21,28)(23,27)(25,32)(29,36)(31,34)(33,35);(1,2)(3,4)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,23,26,36,25,19,34,35)(14,30,24,29,31,27,32,33);(2,4)(9,25)(14,31)(19,23)(24,32)(26,34)(27,30)(29,33)(35,36);(2,4)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,25)(20,22)(24,27)(26,36)(30,32)(31,33)(34,35);(5,9,22,34,11,25,10,26)(6,14,16,32,17,31,15,24)(7,19,21,36,8,23,20,35)(12,27,13,29,28,30,18,33)Polycyclic presentation:
2^9|b^2=f,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426291);